In the definition of submersion you should assume that $d_xf\colon \textrm{T}_x\to \textrm{T}_{f(x)}$ is onto. In this case there is an almost horizontal lift $\phi_x\colon\textrm{T}_{f(x)}\to\textrm{T}_x$ such that $d_xf\circ\phi_x=\mathrm{id}$.

Moreover, applying partition of unity, you can assume that $\phi$ depends continuously on $x$. It makes possible to lift homotopy from $N$ to $M$.

Since $\textrm{diam} F_x$ is small, we can connect any two of its points (say $x$ and $y$) by a short curve. Project this curve to $N$; it is a loop based at $f(x)$. Then shrink the loop to $f(x)$. By lifting this homotopy to $M$ we get that a curve from $x$ to $y$ that lies in the fiber.

(This argument is nearly the same  here)

In the definition of submersion you should assume that $d_xf\colon \textrm{T}_x\to \textrm{T}_{f(x)}$ is onto. In this case there is an almost horizontal lift $\phi_x\colon\textrm{T}_{f(x)}\to\textrm{T}_x$ such that $d_xf\circ\phi_x=\mathrm{id}$.

Moreover, by applying the partition of unity, you can assume that $\phi$ depends continuously on $x$. This makes it possible to lift homotopy from $N$ to $M$.

Since $\textrm{diam}\, F_x$ is small, we can connect any two of its points (say $x$ and $y$) by a short curve. Project this curve to $N$; it is a loop based at $f(x)$. Then shrink the loop to $f(x)$. By lifting this homotopy to $M$, we get a curve from $x$ to $y$ that lies in the fiber.

(This argument is nearly the same as here)